Numerical Methods for Partial Differential Equations Seminar

A metric graph is what one might expect intuitivelya spiderweb, electrical power grid, water-supply system or the British Rail. Physical continuity allows posing one-dimensional differential equations along each edge. Though rather than simple boundary conditions, solutions must satisfy conditions at the vertices which depend on the dynamics along incident edges

We study the eigenvalues and eigenfunctions of the Laplace operator on general metric graphs, which involves solving a system of nonlinear algebraic vertex conditions. In particular, we consider the continuum limit in which graphs become dense within some prescribed embedding space, e.g. a spiderweb filling in the unit disc. As the number of vertices increases, numerical solutions require a novel matrix-determinant root-finding algorithm. Alternatively, in the continuum limit, the discrete system on the vertices reduces to the eigenvalue equation of a contiguous partial differential operator. This resembles in many ways a Laplace-Beltrami operator on a Riemannian manifold but with several notable differences. Rather than a metric tensor, we derive a distinct symmetric tensor that scales differently with distance. Rather than the determinant-based volume form, we find an analogous matrix-trace-based distance form. Our findings open the possibility for a new kind of manifold geometry similar to geodesic structure, but made from underlying “graph material”. We discuss a series of examples of high-density networks, comparing PDE solutions to numerical solutions of the vertex systems. This is joint work with Geoffrey Vasil (U. Edinburgh).

Would Kepler have discovered his laws if machine learning had been around in 1609? Or would he have been satisfied with the accuracy of some black box regression model, leaving Newton without the inspiration to find the law of gravitation? In this talk I will present a review of some industry-oriented machine learning algorithms, and discuss a major issue facing their use in the natural sciences: a lack of interpretability. I will then outline several approaches I have created with collaborators to help address these problems, based largely on a mix of structured deep learning and symbolic methods. This will include an introduction to the PySR/SymbolicRegression.jl software (https://astroautomata.com/PySR), a Python/Julia package for high-performance symbolic regression. I will conclude by demonstrating applications of such techniques and how we may gain new insights from such results.

We consider the problem of "wall-to-wall optimal transport," in which we attempt to maximize the transport of a passive temperature field between hot and cold plates. Specifically, we optimize the choice of the divergence-free velocity field in the advection-diffusion equation amongst all velocities satisfying an enstrophy constraint (which can be understood as a constraint on the power required to generate the flow). Previous work established an a priori upper bound on the transport, scaling as the 1/3-power of the flow's enstrophy. Recently, Doering & Tobasco'19 constructed self-similar two-dimensional steady branching flows saturating this bound up to a logarithmic correction to scaling. This logarithmic correction appears to arise due to a topological obstruction inherent to two-dimensional steady branching flows. We present a construction of three-dimensional ``branching pipe flows" that eliminates the possibility of this logarithmic correction and therefore identifies the optimal scaling as a clean 1/3-power law. Also, using an unsteady "branching blob flow" construction, it appears that the 1/3 scaling is, in fact, optimal in two dimensions as well. We discuss the underlying physical mechanism that makes the branching flows "efficient" in transporting heat and present a design of a mechanical apparatus that can transfer heat close to the best possible scenario.

Many dispersive wave equations have a Hamiltonian structure, with important conserved quantities of both linear and nonlinear type. The accuracy of numerical discretizations over long times depends critically on the preservation of these invariants. I will present a class of invariant-preserving discretizations based on summation-by-parts in space and relaxation in time; these fully-discrete schemes conserve both linear and nonlinear invariants and can be either implicit or explicit. These schemes have been developed for a wide range of such equations, including Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. I will show examples demonstrating that the error for such schemes grows only linearly in time, whereas for general schemes the error grows quadratically. I will also show examples of non-dispersive hyperbolic systems where such invariant-preserving schemes lead to a similar (drastic) improvement in long-time accuracy. Finally, I will present some preliminary results of conserving multiple nonlinear invariants in multi-soliton solutions.

Koopman operators globally linearize nonlinear dynamical systems, and their spectral information can be a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite- dimensional, and computing their spectral information is a considerable challenge. We introduce measure-preserving extended dynamic mode decomposition (mpEDMD), the first Galerkin method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. mpEDMD is a data-driven and structure-preserving algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman operators using a general dictionary of observables. It is flexible and easy to use with any pre-existing DMD-type method and with different data types. We prove the convergence of mpEDMD for projection-valued and scalar-valued spectral measures, spectra, and Koopman mode decompositions. For the case of delay embedding (Krylov subspaces), our results include convergence rates of the approximation of spectral measures as the size of the dictionary increases. We demonstrate mpEDMD on a rang of challenging examples, its increased robustness to noise compared with other DMD-type methods, and its ability to capture the energy conservation and cascade of a turbulent boundary layer flow with Reynolds number > 60000 and state-space dimension >100000. Finally, if time permits, we discuss how this algorithm forms part of a broader program on the foundations of infinite-dimensional spectral computations.

Inverse design is the direct optimization of a target property, it has the potential to automatize design for real-world engineering problems. However, inverse design is limited by the simulation capabilities of physical phenomena. In an iterative optimization process, a computer model of a real-world phenomenon is queried hundreds to thousands of times. On the one hand, in many applications, there exist physical models where the simulations are accurate enough to result in a meaningful design, but these simulations may be too resource-intensive to run the optimization process. This is the case for example in metasurface designoptical devices that present both subwavelength aperiodic patterns and a thousands-of-wavelengths-long diameter. On the other hand, data-driven modls are most often very fast to evaluate and they do not require a complete knowledge of the process being optimized. Unfortunately, as the number of real-world design parameters increases, these models may require unreasonable amounts of data and resources to be trained accurately. In this talk, I will present methodologies that combine data and models to accelerate simulations and enable inverse design. The center-piece of these approaches is the creation of data- and resource-efficient global surrogate models that are repeatedly called in the optimization loop. I will show theory and experimental results of surrogate-based large-scale metasurface designs. I will also show recent work in active learning and scientific machine learning (SciML) for various engineering problems, where information from both physical models and data are optimally leveraged to achieve efficient inverse design. Upon inspection, SciML may also give insights and intuition about the engineering process being simulated and optimized.